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Full text of "Treatise On Analysis Vol-Ii"

7   SEMICONT1NUOUS FUNCTIONS       25

(12.7.4)    A subset A of a topological space E is open (resp, closed) in E if and
only if (pA is lower (resp. tipper) semicontinuous on E.

This follows immediately from (12.7.2).

(12.7.5)    Let f> g be two mappings of a topological space E into R, each of
which is lower semicontinuous at a point XQ e E. Then the functions sup(/, g)
andmf(f, g) are lower semicontinuous at x0. The same is true for /+ g if the
sumf(x) + g(x) is defined for all x e E (4.1.8), and for fg if f and g are both
^0 and the product f(x)g(x) is defined for allxeE (4.1.9).

We shall give the proof for f+g\ the other cases are analogous. The
result is obvious if f(x0) or #(*o) *s equal to — oo. If not, then we have
f(x0)+g(x0)> — oo. Every number a e R such that a <f(x0) + #(*o) can be
written in the form a = /? + y, with /? <f(x0) and y < g(xQ) (it is enough to
choose y such that a —f(x0) <y < g(x0)). By hypothesis, there exists a neigh-
borhood V of x0 such that, for all x e V, we have p <f(x) and y < g(x).
It follows that a = /? + y <f(x) + #(;c) for all x e V. Hence the result.

If/is /0wr semicontinuous at a point JCG, and if/^ 0 on E, then the
function I//(where l/f(x) is taken to be +00 if f(x) = 0) is upper semi-
continuous at x0. The proof is analogous.

Given a set E and any family (/a)a e, of mappings of E into E, the upper
(resp. lower) envelope of the family is defined to be the mapping xh->sup/a(x)

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(resp. .xi—nnf fa(x)) of E into 5. It is denoted by sup/a (resp. inf^). We

ae I                                                                   ael                  ael

have sup (-/«) = -inf/a.

ae I                    ael

(12.7.6)    Let E be a topological space and let (>/a)a 6 r Z?e a family of mappings of
E mfo K. If each fa is lower semicontinuous at a point x0 e E, then the upper
envelope f= sup/a is lower semicontinuous at x0.

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Given any A </(^0) = suP/a(:!Co)J there exists by hypothesis an index

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]8 e I such that A <fp(x0). Since/^ is lower semicontinuous at the point x0,
there exists a neighborhood V of x0 such that 1 <fp(x) for all xe V, and
therefore A </(*) for all x e V.
In particular:

(12.7.7)    The upper (resp. lower) envelope of a family of continuous mappings of
E into K is lower (resp. upper) semicontinuous. Ak (1 ^ k g m).e Bk for some k ^ n. Hence the sets BM cover E.